where \(\mathcal {Q}\)  is a set of distributions close to the empirical training distribution \(\mathbb {P}_n\)  that we call the uncertainty set. The choice of this \(\mathcal {Q}\)  establishes how the training distribution can be perturbed, with most techniques choosing all the distributions such that a f-divergence [1]}, [2]} or a Wasserstein distance [3]} is smaller than a certain threshold, or by modeling it with a generative network [4]}. If some causal information is available and if, so called interventions
on the sensitive variable, can be modeled as distributional shifts on the distributions, hence distributional robust models with respect to such shifts will be protected from the bias induced by this variable.
Therefore, distributional robustness extends stability requirements to achieve fairness by controlling the output of the algorithm in the worst distributional case around the observed empirical distribution.
